Eddy current tomography: A Bayesian approach with a compound weak membrane-Beta prior model by O. Venard1,3, D. Pre´mel1 and A. Mohammad-Djafari2 1
LESiR, URA CNRS 1375, ENS Cachan, 94235 Cachan, France; 2 LSS, UMR CNRS-Supe´lec-UPS 014, ESE, 91192 Gif sur yvette, France; 3 LSM, ESIEE, 93162 Noisy le grand, France Abstract This work deals with an Eddy current imaging system. The solution of the associated inverse problem is regularized in a Bayesian estimation framework. The proposed approach combines the following a priori information: the function to be reconstructed had to be piecewise continous and is bounded between and . This compound a priori knowledge allows us to enhance reconstruction results with respect to the shape precision and the size resolution of the objects to be reconstructed in this kind of application. Keywords: Eddy current tomography, Inverse problem, Regularization, Bayesian estimation, Beta law, Weak membrane, Markov random field.
1.
Introduction
Among Non-Destructive Evaluation techniques (NDE), eddy current tomography aims to image surface or near surface flaws in electrically conductive material. The goal is thus to produce a spatial map (object function or flaw function) of the flaw from the measured data. The modifications to the eddy currents caused by the flaw give rise to change in the scattered magnetic field. A coil is used to induce eddy currents and we sample the component of the scattered magnetic field , normal to the air-conductor interface thanks to a small coil. One difficulty of this inverse problem comes from the non-linearity of the relationship between the object function and the observed data. The scattered magnetic field is related to the flaw function by two coupled integral equations (the forward problem). The first one computes the electric field diffracted by the flaw and the second one gives the normal component of the scattered magnetic field given the diffracted electric field. Different approaches have been proposed to solve this non linear inverse problem: one can cite the iterative linearization method [6] and the simultaneous reconstruction of the internal diffracted field and of the object function [4, 5]. This inverse problem may be linearized under some hypotheses using the B ORN approximation. In this work, we assume those hypotheses to be fullfilled [11, 14]. The sensor response is then linked to the flaw, uniformly illuminated by the incident electric field, through an infinite
[3], which models the transfer function of the experimental system. half space G REEN dyadic function This linear inverse problem is then the solution of a F REDHOLM integral equation of the first kind. This is an inherently ill-posed inverse problem [7, 8], which must be regularized to yield a stable and physically plausible solution. The classical smoothness constraint on the solution [1, 8], does not allow recovery of the discontinuities of the original object function. In our case, the observed medium can be made of several patches of different conductivity separated from each other by abrupt contours. This strong local correlation between pixels, apart from those close to the discontinuities, suggests to represent the object (the relative conductivity ) by a binary, piecewise continous function (the weak membrane model)[2]. When the object to be reconstructed is mainly the a priori knowledge introduced by the Beta law [15] enables us to restrict the admissibles values for each pixel to the interval . However this a priori does not involve any correlation between pixels. Some authors have proposed to add a functional modeling a binary field to the "weak membrane" in the regularization process [13]. The
corresponding Maximum a posteriori (MAP) criterion is then minimized thanks to an optimization algorithm called Graduated Non-convexity (GNC) [2]. In this communication we propose to combine the weak membrane model with a Beta law. This allows us to work with real valued variables and at the same time to obtain a solution whose distribution is concentrated near and without excluding intermediate values. This paper is structured as follows: in section 2 we recall the forward problem statement in the context of the B ORN approximation. After settling the inverse problem in section 3, we discuss the choice of regularizing functional depending on the information we want to gather in the a priori knowledge. This section ends with some guidelines to the associated optimization problem. Finally we present some simulation results illustrating the performances of the proposed method. 2.
The forward problem We consider the experimental system model [11] showed in Figure 1. The solution of the forward problem :9 24365 7 8 $&8 7 .1* ')(+*-,/.10 A@CBD> !#" $&% Figure 1. Simulated system
linearized using B ORN approximation is expressed as follow: FEHJG I LK&MONQPRP
SUT
H G X1Y V Z H G X[Y \ G P V ] H G X^Y`_ a HG X
#W V HG Q
(1)
# EcH G H G X I ed #fhgi #fhjCi Jk [3] relates an electrical source in region 2 ( l`mn ) located at o X EcH G X I to where b b b the normal component of the scattered magnetic field at an observation point o EcHJG I in region 1 ( l`pn ). The vector \G P stands for the induced excitation field without flaw which can be computed thanks to the method proposed by Luquire et al. [10]. frq i
sct The object function EcH G X I C is the relative variation of the conductivity in the investigated domain u , highlighted in Figure 1. R P is the healthy medium conductivity in the domain u and Rv EcH G X I is the conductivity in the flawed region.
After discretisation by Moment Method [9] the equation (1) becomes: w Erx yz l P I | {AMON P R P} g } j~)
~
t g th j t
\ # W V x|x X Cy y X l ^X Y V x X Cy X l X^Y G P V x X y X l X^Y
(2)
where
V x
x X y
S Q
} g } j sinc Q # I ^ f g g i ^f j j t i _ _ t t Gx Gy } K E sinh ¡
y X l X YFa Gx
Gy
JGl C
sinc D
¦ |£ and £ ¤{UN P R P M and where and are the F OURIER with ¢ , domain variables. The g j domain u surrounding the flaw is discretised uniformly in voxels of size ¥ } } } .
3.
The inverse problem After discretisation the problem becomes the resolution of the following equation : §
C® « C¯A°²±J³ Eµ´¶ Iz/· ¹¸ Lº n¨ª©¬
· |¸O» ¸
where · is a vector containing the values of the object function on the discretised observed domain u and stands C® « is computed for the noise, assumed additive, coming from discretisation and measurement errors. The matrix ¨ ©¬
f^gi E C l X I and #¼f^jCi E C Cl X I . from the impulse responses
#fhgi is displayed Figure 2(a), a The modulus of the F OURIER transform (FT) of the discretised operator b slice vue of this figure for is shown Figure 2(b). This transform could be seen as an approximation of the spectral decomposition in the case of a convolutional operator. The bandpass behaviour of this operator highlights
# smoothes the discontinuities of the object the ill-posedness of the inverse problem stated by (1). The operator b and cancels the information about continous region. Because of the ill-posedness of the continous problem, the discrete operator º is ill-conditionned. The ill-conditionned character of the discrete problem is degraded as the discretisation step } g or } j becomes smaller. 4
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Figure 2.
Regularization in a Bayesian framework allows us to introduce a priori knowledge about the object function as a prior probability distribution function and therefore fills up the lack of information in the measurements. In this approach we define the solution as the MAP estimate : É Ê
±J˳cÌw°1Í I I Î Ï E with Ï E
ÑÐ Í EµÒQE ÓÕÔ I#I
where ÒaE ÓÕÔ I is the posterior law given by ÒQE ÓÕÔ I
ÒQE Ô?Ó/ I Q Ò E I ÒQE Ô I
If the noise is assumed zero mean, white and Gaussian with known variance then Ï E I becomes: Û
IÖØ× Ô WÙ × ÚQÛ E I Ï E
where E I is the energy function of the prior law. The originality of our method is in the choice of an appropriate prior model for Eddy current tomography application. 3.1. Choice of the a priori law 3.1.1. Object function support The object function we are looking for is the relative conductivity ÜÝ CÞ , where K is the pixel index and RÜ the flaw region and equal to in the its associated conductivity. When the flaw is a void E RÜ?ß I , Ü is equal to in I healthy region E RÜ?ßR P of the investigated domain u . Thus it will take mostly one of this both value, the others ones belonging to the interval [ c are less likely. Therefore its support is restricted to the interval Õ à . This strong a priori knowledge can be translated by a Beta law (Figure 3(a)). ÒQE Ü IOÖ
Üz á E
Ü I â È Üã
äà ä and ä Cåæ #ç
p
If the pixels are assumed independent, the related energy can be written [14, 15]: Û c E· I
ÑÐ Í ÒQE ·cI Lèé#ê å ~ Ü
Ð Í E Ü I
ç ~ Ü
Ð Í E Ü I
However, such a choice has a drawback. The MAP criterion or equivalently the energy related to this posterior law ·cI Ï E
× § º ·z× ßÛ E ·AI
(3)
goes to ¬ë for ÜFØ or ÜÆ . We must therefore restrict the object function support to the interval ä ä . The reconstruction results could then become more sensitive to ä than to å and ç . To remedy to this situation, one can choose a law of the same kind (Figure 3(b)), well defined on with maxima close to and such as: ÑÐ Í ÒZE ·cI Lèéê ~ Ü
å? Ü Ð Í E å? Ü I ~
Ü
ç E Ü I Ð Í E ç E
Ü I#I
3.1.2. Local correlation property of the flaw function The a priori information involved by the Beta law does not take into account the local correlation between pixels. Whereas the relative conductivity of one pixel is very likely to be the same as those of its neighbours excepted the boundaries of the flaw. Therefore it could be fruitfull to introduce this a priori knowledge by means of a piecewise continous function. The energy related to this a priori model is: Û c E· I
~ Ü
Ôzì í E Ü
Ü I
with Ô ì í µE î I ð z ï N
Ú î
í
if Ó î ÓUmòñ ì » otherwise
(4)
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óàôöõh÷ ø^ù ú¼ôöõ^÷ ø ûÕûÕû : óàôöõh÷ ü^ù ú¼ôöõ^÷ ý - û : óàôcõ^÷ ý^ù ú¼ôöõ^÷ ü
1.3
-:
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óàôcõ^÷ øhù ú¼ôöõh÷ ø ûÕûÕû : óàôcõ^÷ þhù ú¼ôöõh÷ ÿ - û : óôöõ^÷ ÿhù ú¼ôöõh÷ þ -:
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Figure 3.
Ú
An implicit edge process is involved in the regularizing functional Ôvì í E [I , where acts as a scale parameter which gives a constraint on the size of the homogeneous patches we seek to reconstruct. The parameter N is given Ú , where Ô P is the threshold above which a discontinuity is introduced. This regularizing functional by Ô P N is not convexe, and the associated optimization problem can be solved with Graduated Non Convexity (GNC) algorithm. This kind of Markovian regularization functional and its associated optimization algorithm above-mentioned are further detailled in [2, 12, 13].
3.1.3. Compound criterion Û
Û
Û
We proposed two regularization functions E I and E I , each of them favoring different properties: E I Û permits us to enforce the distribution of the solution of the inverse problem to be concentrated near to and . E I permits us to enforce the solution to have homogeneous patches. We propose then to combine these two functionals to obtain a satisfactory solution. Thus we define the solution as the minimizer of the following criterion: ·cI Ï E
× § º ·z× ßÛ E ·AI ¤Û E ·AI »
(5)
This criterion allows us to introduce all of the a priori knowledge we are able to express concerning the physical aspect of the flawed region. 3.2. Implemented optimization algorithm Neither the weak membrane model nor the Beta law energies are convexe functions. Consequently, we need a global optimization technique to reach the solution. Between these techniques we can mention the simulated annealing. But its calculation cost is in general huge and more specifically in our case, where the support of the operator º is large. Recently some authors have shown that the continuation methods, such as GNC can be used in these situations to try to obtain a solution close to the global one [12]. In the sequel we will see that the optimization problem involved by the minimization of (5) can be achieved providing some fairly precautions. Û
To overcome the minimization of the energy E I , care must be taken to choose the initialization step far enough from the boundaries of the object function support ( ä or ä ). A safely choice could be a uniform 0.5 valued initial object function. To show that this choice is not very critical, we have chosen a uniform 0.05 initial step for all the subsequent simulation whose results are presented in the next section. To explain this we can observe that during the first step of the GNC algorithm, the Beta energy remains quite low and it starts only to bias the solution after some iterations as soon as some values of the object function get closer to the support boundaries.
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4.
Simulation results and conclusion
In this section we give some simulation results to show the behaviour and the efficiency of the different regularization criteria described in the above section. The data are simulated using (2) with a test object (Figure 4(a)) consisting of a cluster of three closed flaws. We added a white Gaussian noise to these data to simulate measurement
noise and modeling errors. The SNR is equal to _ (Figure 4(c)). The modulus of the impulse response of the » » I , is represented Figure 4(b).
#fhgi for a discretisation grid E Ê kernel b
The least square (LS) solution obtained by a conjugated gradient algorithm is shown in Figure 5(a). When we use the same criterion with a stopping rule (Figure 5(b)), the obtained result is characterized by its smoothness and its low contrast. The Beta law regularization (Figure 5(c)) allows to obtain a highly contrasted image, even if the result is not perfect. The weak membrane model (Figure 5(d)) tends to gather neighbouring pixels. In our case it cannot resolve two close flaws. Finally, the proposed joint regularizing function (5) preserves details while building homogeneous patterns (Figure 5(e)). In this example it leads to perfect reconstruction. Those results appear to be promising and we plan to use the proposed method for the reconstruction of buried flaws. We also hope to reduce the size of the flaws to be reconstructed. An experimental system, similar to the one depicted in Figure 1 will soon allow us to validate this Eddy current imaging system with real data.
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